Find the mass of the lamina described by the inequalities $x \geq 0$ and $3 \leq y \leq 3 + \sqrt { 9 - x ^ { 2 } }$, given that its density is $\rho ( x , y ) = 7 x y$. (Hint: Some of the integrals are simpler in polar coordinates.)
A) $\frac { 2,079 } { 8 }$
B) $\frac { 16 } { 6,237 }$
C) $\frac { 8 } { 2,079 }$
D) $\frac { 2,457 } { 8 }$
E) $\frac { 6,237 } { 16 }$

Question

Quizplus · Accepted Answer

To find the mass of the lamina, we integrate the density function over the given region. The region described by $3 \leq y \leq 3 + \sqrt { 9 - x ^ { 2 } }$ and $x \geq 0$ is a semicircle with radius 3 centered at $(0,3)$ on the right side of the y-axis. Converting to polar coordinates, where $x = r\cos(\theta)$ and $y = r\sin(\theta)$, the density function becomes $\rho(r, \theta) = 7r^2\cos(\theta)\sin(\theta)$. The limits for $r$ are from 0 to 3, and for $\theta$, from 0 to $\frac{\pi}{2}$ to cover the semicircle's area. The integral in polar coordinates is:$$\int_{0}^{\frac{\pi}{2}}\int_{0}^{3} 7r^2\cos(\theta)\sin(\theta) \, drd\theta$$Solving this double integral gives the mass of the lamina as $\frac { 2,079 } { 8 }$.

Find the Mass of the Lamina Described by the Inequalities x≥0x \geq 0x≥0

Find the Mass of the Lamina Described by the Inequalities $x \geq 0$